Lie Groups Of Rank Research Articles

The rates of growth in an acylindrically hyperbolic group

Let G be an acylindrically hyperbolic group on a \delta -hyperbolic space X . Assume there exists M such that for any finite generating set S of G , the set S^{M} contains a hyperbolic element on X . Suppose that G is equationally Noetherian. Then we show the set of the growth rates of G is well ordered. The conclusion was known for hyperbolic groups, and this is a generalization. Our result applies to all lattices in simple Lie groups of rank 1, and more generally, relatively hyperbolic groups under some assumption. It also applies to the fundamental group, of exponential growth, of a closed orientable 3 -manifold except for the case that the manifold has Sol-geometry.

On absolutely profinitely solitary lattices in higher rank Lie groups

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp ⁡ ( n , R ) \operatorname {Sp}(n,\mathbb {R}) , G 2 ( 2 ) G_{2(2)} , E 8 ( C ) E_8(\mathbb {C}) , F 4 ( C ) F_4(\mathbb {C}) , and G 2 ( C ) G_2(\mathbb {C}) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Abnormal Extremals in the Sub-Riemannian Problem with Growth Vector $(2,3,5,8,14)$

We study the left-invariant sub-Riemannian problem on the free nilpotent Lie group of rank $2$ and step $5$. We describe some abnormal trajectories and some properties of the set filled by nice abnormal trajectories starting at the identity of the group.

Correction: Global Stability of the Pluriclosed Flow on Compact Simply Connected Simple Lie Groups of Rank Two

A Correction to this paper has been published: https://doi.org/10.1007/s00031-022-09777-x

Global Stability of the Pluriclosed Flow on Compact Simply Connected Simple Lie Groups of Rank Two

AbstractWe compute the (1,1)-Aeppli cohomology of compact simply connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott–Chern and the Aeppli cohomologies, arises from just the left-invariant forms and we computed the whole Bott–Chern diamonds of SU(3) and Spin(5) when they are equipped with a left-invariant isotropic complex structure.

On the profinite rigidity of lattices in higher rank Lie groups

AbstractWe investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$ , $F_4$ , and $G_2$ . In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\textrm{SL}_{2n+1}(\mathbb{R})$ , $\textrm{SL}_{2n+1}(\mathbb{C})$ , $\textrm{SL}_n(\mathbb{H})$ , or groups of type $E_6$ .

Geodesic orbit spaces of compact Lie groups of rank two

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H, g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

Cohomology of the spaces of commuting elements in Lie groups of rank two

Let G be a compact, connected Lie group, and let Hom(Zm,G) denote the space of commuting m-tuples in G. Baird proved that the cohomology of Hom(Zm,G) is identified with a certain ring of invariants of the Weyl group of G. In this paper by using the result of Baird we give the cohomology ring of Hom(Z2,G) for simple, simply connected Lie groups G of rank 2.

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector (2,3,5). In previous works, we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time — the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work, we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.

The Spectrum of the Laplace Operator on Connected Compact Simple Lie Groups of Rank Four. II

The Spectrum of the Laplace Operator on Connected Compact Simple Lie Groups of Rank Four. II

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.

Connes-Landi spheres are homogeneous spaces

In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra.

The geometry of maximal representations of surface groups into SO0(2,n)

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

Fock-Goncharov coordinates for rank two Lie groups

Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for the flip (changing the diagonal in a quadrilateral). This gives explicit coordinates on higher Teichmuller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in $${{\,\mathrm{Sp}\,}}(4,{\mathbb {C}})/\langle -I\rangle $$ for the figure-eight knot complement.

Simplicial volume, barycenter method, and bounded cohomology

We prove that the locally finite simplicial volume of most \({\mathbb {Q}}\)-rank 1 locally symmetric spaces is positive, which has been open for many years. As a corollary, we improve the degree theorem of Connell and Farb for \({\mathbb {Q}}\)-rank 1 locally symmetric spaces. During the course of a proof, we show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most higher rank symmetric spaces. We also address the issue of surjectivity of the comparison map for semisimple Lie groups of rank 2.

On Mimura's extension problem

We determine the group structure of the 23-rd homotopy group π23(G2), where G2 is the exceptional Lie group of rank 2, which hasn't been determined for 50 years.

The odd primary order of the commutator on low rank Lie groups

Let G be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime p. After localization at p, there is a space A which “generates” G in a certain sense. Assuming G satisfies a homotopy nilpotency condition relative to p, we show that the Samelson product 〈1G,1G〉 of the identity of G equals the order of the Samelson product 〈ı,ı〉 of the inclusion ı:A→G. Applying this result, we calculate the orders of 〈1G,1G〉 for all p-regular Lie groups and give bounds of the orders of 〈1G,1G〉 for certain quasi-p-regular Lie groups.

Phase portraits of the full symmetric Toda systems on rank-2 groups

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,ℝ) and the real form of G2.

The spectrum of the Laplace operator on connected compact simple Lie groups of rank four

In the present article, we explicitly compute the spectrum of the Laplace operator on smooth real-valued and complex-valued functions on connected compact simple Lie groups of rank four with a bi-invariant Riemannian metrics that correspond to the root systems B 4, C 4, and D 4. We also find a connection between the obtained formulas, number theory, and integral quadratic forms in two, three, and four variables.